Griffiths R.B. Consistent Quantum Theory

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Unitary dynamics

7.1 The Schr

odinger equation

The equations of motion of classical Hamiltonian dynamics are of the form

∂ H

∂p

=−

∂ H

∂x

, (7.1)

where x

, x

, etc. are the (generalized) coordinates, p

, p

, etc. their conjugate

momenta, and H(x

, p

, x

, p

,...) is the Hamiltonian function on the classical

phase space.

In the case of a particle moving in one dimension, there is only a single coordi-

nate x and a single momentum p, and the Hamiltonian is the total energy

H = p

/2m + V(x), (7.2)

with V(x) the potential energy. The two equations of motion are then:

dx/dt = p/m, dp/dt =−dV/dx. (7.3)

For a harmonic oscillator V(x) is

, and the general solution of (7.3) is given

in (2.1), where ω =

√

K/m.

The set of equations (7.1) is deterministic in that there is a unique trajectory or

orbit γ(t) in the phase space as a function of time which passes through γ

at t = 0.

Of course, the orbit is also determined by giving the point in phase space through

which it passes at some time other than t = 0. The orbit for a harmonic oscillator

is an ellipse in the phase plane; see Fig. 2.1 on page 12.

The quantum analog of (7.1) is Schr

odinger’s equation, which in Dirac notation

can be written as

|ψ

=H|ψ

, (7.4)

where H is the quantum Hamiltonian for the system, a Hermitian operator which

7.1 The Schr

odinger equation 95

may itself depend upon the time. This is a linear equation, so that if |φ

 and |ω



are any two solutions, the linear combination

|χ

=α|φ

+β|ω

 (7.5)

is also a solution, where α and β are arbitrary (time-independent) constants. Equa-

tion (7.4) is deterministic in the same sense as (7.1): a given |ψ

 at t = 0gives

rise to a unique solution |ψ

 for all values of t. The result is a unitary dynamics

for the quantum system in a sense made precise in Sec. 7.3.

The Hamiltonian H in (7.4) must be an operator deﬁned on the Hilbert space

H of the system one is interested in. This will be true for an isolated system, one

which does not interact with anything else — imagine something inside a com-

pletely impermeable box. It will also be true if the interaction of the system with

the outside world can be approximated by an operator acting only on H. For ex-

ample, the system may be located in an external magnetic ﬁeld which is effectively

“classical”, that is, does not have to be assigned its own quantum mechanical de-

grees of freedom, and thus enters the Hamiltonian H simply as a parameter.

One is sometimes interested in the dynamics of an open (in contrast to isolated)

subsystem A of a composite system A ⊗ B when there is a signiﬁcant interaction

between A and B. Of course (7.4) can be applied to the total composite system,

assuming that it is isolated. However, there is no comparable equation for the sub-

system A, as it cannot, at least in general, be described by its own wave function,

and its dynamical evolution is inﬂuenced by that of the other subsystem B, often

referred to as the environment of A. Constructing dynamical equations for open

subsystems is a topic which lies outside the scope of this book, although Ch. 15

on density matrices provides some preliminary hints on how to think about open

subsystems.

For a particle in one dimension moving in a potential V(x), (7.4) is equivalent

to the partial differential equation

∂ψ

∂t

=−

∂

∂x

+ V(x)ψ (7.6)

for a wave packet ψ(x, t) which depends upon the time as well as the position

variable x. The Hamiltonian in this case is the linear differential operator

H =−

∂

∂x

+ V(x). (7.7)

In general it is much more difﬁcult to ﬁnd solutions to (7.6) than it is to integrate

(7.3). A formal solution to (7.6) for a harmonic oscillator is given in (7.23).

One way to think about (7.4) is to choose an orthonormal basis {|j, j =

1, 2,...} of the Hilbert space H which is independent of the time t. Then (7.4)

96 Unitary dynamics

is equivalent to a set of ordinary differential equations, one for each j:

j|ψ

=j|H|ψ

. (7.8)

This is somewhat less abstract than (7.4), because both j|ψ

 and j|H|ψ

 are

simply complex numbers which depend upon t, and thus are complex-valued func-

tions of the time. By writing |ψ

 as a linear combination of the basis vectors,

|ψ

=



|jj|ψ

=



(t)|j, (7.9)

with time-dependent coefﬁcients c

(t), and expressing the right side of (7.8) in the

form

j|H|ψ

=



j|H|kk|ψ

, (7.10)

one ﬁnds that the Schr

odinger equation is equivalent to a collection of coupled

linear differential equations

hdc

/dt =



j|H|kc

(7.11)

for the c

(t). The operator H, and therefore also its matrix elements, can be a

function of the time, but it must be a Hermitian operator at every time, that is, for

any j and k,

j|H(t)|k=k|H(t)|j

∗

. (7.12)

When H is two-dimensional, (7.11) has the form

hdc

/dt =1|H|1c

+1|H|2c

hdc

/dt =2|H|1c

+2|H|2c

(7.13)

These are linear equations, and if H, and thus its matrix elements, is independent of

time, one can ﬁnd the general solution by diagonalizing the matrix of coefﬁcients

on the right side. Let us assume that this has already been done, since we earlier

made no assumptions about the basis {|j}, apart from the fact that it is independent

of time. That is, assume that

H = E

|11|+E

|22|, (7.14)

so that the off-diagonal terms 1|H|2 and 2|H|1 in (7.13) vanish, while the

diagonal terms are E

and E

. Then the general solution of (7.13) is of the form

(t) = b

−iE

, c

(t) = b

−iE

, (7.15)

where b

and b

are arbitrary (complex) constants.

7.1 The Schr

odinger equation 97

The precession of the spin angular momentum of a spin-half particle placed

in a constant magnetic ﬁeld is an example of a two-level system with a time-

independent Hamiltonian. If the magnetic ﬁeld is B = (B

, B

), the Hamilto-

nian is

H =−γ(B

+ B

), (7.16)

where the spin operators S

, etc., are deﬁned in the manner indicated in (5.30), and

γ is the gyromagnetic ratio of the particle in suitable units. This Hamiltonian will

be diagonal in the basis |w

, |w

−

, see (4.14), where w is in the direction of the

magnetic ﬁeld B.

The preceding example has an obvious generalization to the case in which a

time-independent Hamiltonian is diagonal in an orthonormal basis {|e

}:

H =



e

|. (7.17)

Then a general solution of Schr

odinger’s equation has the form

|ψ

=



−iE

, (7.18)

where the b

are complex constants. One can check this by evaluating the time

derivative

|ψ

=



−iE

, (7.19)

and verifying that it is equal to H|ψ

. An alternative way of writing (7.18) is

|ψ

=e

−itH/

|ψ

, (7.20)

where |ψ

 is |ψ

 when t = 0. The operator e

−itH/

is deﬁned in the manner

indicated in Sec. 3.10, see (3.97). It can be written down explicitly as

−itH/



−iE

e

|. (7.21)

In the case of a harmonic oscillator, with

= (n + 1/2)

hω (7.22)

the energy and |φ

 the eigenstate of the nth level, (7.18) is equivalent to

ψ(x, t) = (e

−iωt/2

)



−inωt

(x), (7.23)

where φ

(x) is the position wave function corresponding to the ket |φ

.

98 Unitary dynamics

A particular case of (7.18) is that in which b

= δ

, that is, all except one of the

vanish, so that

|ψ

=e

−iE

. (7.24)

The only time dependence comes in the phase factor, but since two states which

differ by a phase factor have exactly the same physical signiﬁcance, a quantum state

with a precisely deﬁned energy, known as a stationary state, represents a physical

situation which is completely independent of time. By contrast, a classical system

with a precisely deﬁned energy will typically have a nontrivial time dependence;

e.g., a harmonic oscillator tracing out an ellipse in the classical phase plane.

The inner product ω

|ψ

 of any two solutions of Schr

odinger’s equation is

independent of time. Thus |ψ

 satisﬁes (7.8), while the complex conjugate of this

equation with ψ

replaced by ω

−i

ω

|j=ω

|H|j, (7.25)

where H

†

on the right side has been replaced with H, since the Hamiltonian (which

could depend upon the time) is Hermitian. Using (7.8) along with (7.25), one

arrives at the result

ω

|ψ

=i



ω

|jj|ψ





ω

|jj|H|ψ

−



ω

|H|jj|ψ

=0, (7.26)

since both of the last two sums are equal to ω

|H|ψ

. This means, in particular,

that the norm ψ

 of a solution |ψ

 of Schr

odinger’s equation is independent of

time, since it is the square root of the inner product of the ket with itself.

The fact that the Schr

odinger equation preserves inner products and norms means

that its action on the ket vectors in the Hilbert space is analogous to rigidly rotating

a collection of vectors in ordinary three-dimensional space about the origin. If one

thinks of these vectors as arrows directed outwards from the origin, the rotation

will leave the lengths of the vectors and the angles between them, and hence the

dot product of any two of them, unchanged, in the same way that inner products

of vectors in the Hilbert space are left unchanged by the Schr

odinger equation. An

operator on the Hilbert space which leaves inner products unchanged is called an

isometry. If, in addition, it maps the space onto itself, it is a unitary operator.

Some important properties of unitary operators are stated in the next section, and

we shall return to the topic of time development in Sec. 7.3.

7.2 Unitary operators 99

7.2 Unitary operators

An operator U on a Hilbert space H is said to be unitary provided (i) it is an

isometry, and (ii) it maps H onto itself. An isometry preserves inner products, so

condition (i) is equivalent to



U|ω



†

U|φ=ω|U

†

U|φ (7.27)

for any pair of kets |ω and |φ in H, and this in turn will be true if and only if

†

U = I. (7.28)

Condition (ii) means that given any |η in H, there is some |ψ in H such that

|η=U|ψ. This will be the case if, in addition to (7.28),

†

= I. (7.29)

The two equalities (7.28) and (7.29) tell us that U

†

is the same as the inverse U

−1

the operator U. For a ﬁnite-dimensional Hilbert space, condition (ii) for a unitary

operator is automatically satisﬁed in the case of an isometry, so (7.28) implies

(7.29), and vice versa, and it sufﬁces to check one or the other in order to show that

U is unitary.

If U is unitary, then so is U

†

. In addition, the operator product of two or more

unitary operators is a unitary operator. This follows at once from (7.28) and (7.29)

and the rule giving the adjoint of a product of operators, (3.32). Thus if both U and

V satisfy (7.28), so does their product,

(UV)

†

UV = V

†

UV = V

†

IV = I, (7.30)

and the same is true for (7.29).

A second, equivalent deﬁnition of a unitary operator is the following: Let {|j}

be some orthonormal basis of H. Then U is unitary if and only if {U|j} is also an

orthonormal basis. If H is of ﬁnite dimension, one need only check that {U|j} is

an orthonormal collection, for then it will also be an orthonormal basis, given that

{|j} is such a basis.

The matrix {j|U|k} of a unitary operator in an orthonormal basis can be

thought of as a collection of column vectors which are normalized and mutually

orthogonal, a result which follows at once from (7.28) and the usual rule for matrix

multiplication. Similarly, (7.29) tells one that the row vectors which make up this

matrix are normalized and mutually orthogonal. Any 2 × 2 unitary matrix can be

written in the form

iφ



αβ

−β

∗



, (7.31)

100 Unitary dynamics

where α and β are complex numbers satisfying

|α|

+|β|

= 1 (7.32)

and φ is an arbitrary phase. It is obvious from (7.32) that the two column vectors

making up this matrix are normalized, and their orthogonality is easily checked.

The same is true of the two row vectors.

Given a unitary operator, one can ﬁnd an orthonormal basis {|u

} in which it

can be written in diagonal form

U =



u

|, (7.33)

where the eigenvalues λ

of U are complex numbers with |λ

|=1. Just as Hermi-

tian operators can be thought of as somewhat analogous to real numbers, since their

eigenvalues are real, unitary operators are analogous to complex numbers of unit

modulus. (In an inﬁnite-dimensional space the sum in (7.33) may have to be re-

placed by an appropriate integral.) As in the case of Hermitian operators, Sec. 3.7,

if some of the eigenvalues in (7.33) are degenerate, the sum can be rewritten in the

form

U =





, (7.34)

where the S

are projectors which form a decomposition of the identity, and λ



= λ



for k = l.

All operators in a collection {U, V, W,...} of unitary operators which commute

with each other can be simultaneously diagonalized using a single orthonormal

basis. That is, there is some basis {|u

} in which U, V, W, and so forth can

be expressed using the same collection of dyads |u

u

|, as in (7.33), but with

different eigenvalues for the different operators. If one writes down expressions

of the form (7.34) for V, W, etc., the decompositions of the identity need not be

identical with the {S

} appropriate for U, but the different decompositions will all

be compatible in the sense that the projectors will all commute with one another.

7.3 Time development operators

Consider integrating Schr

odinger’s equation from time t = 0tot = τ starting

from an arbitrary initial state |ψ

. Because the equation is linear, the dependence

of the state |ψ

 at time τ upon the initial state |ψ

 can be written in the form

|ψ

=T(τ, 0)|ψ

, (7.35)

where T(τ, 0) is a linear operator. And because Schr

odinger’s equation preserves

inner products, (7.26), T(τ, 0) is an isometry. In addition, it maps H onto itself,

7.3 Time development operators 101

because if |η is any ket in H, we can treat it as a “ﬁnal” condition at time τ and

integrate Schr

odinger’s equation backwards to time 0 in order to obtain a ket |ζ 

such that |η=T(τ, 0)|ζ . Therefore T(τ, 0) is a unitary operator, since it satisﬁes

the conditions given Sec. 7.2.

Of course there is nothing special about the times 0 and τ , and the same argument

could be applied equally well to the integration of Schr

odinger’s equation between

two arbitrary times t



and t, where t can be earlier or later than t



. That is to say,

there is a collection of unitary time development operators T(t, t



), labeled by the

two times t and t



, such that if |ψ

 is any solution of Schr

odinger’s equation, then

|ψ

=T(t, t



)|ψ



. (7.36)

These time development operators satisfy a set of fairly obvious conditions. First,

if t



is equal to t,

T(t, t) = I. (7.37)

Next, since

|ψ

=T(t, t



)|ψ



=T(t, t



)|ψ



=T(t, t



)T(t



, t



)|ψ



, (7.38)

it follows that

T(t, t



)T(t



, t



) = T(t, t



) (7.39)

for any three times t, t



, t



. In particular, if we set t



= t in this expression and use

(7.37), the result is

T(t, t



)T(t



, t) = I. (7.40)

Since T(t, t



) is a unitary operator, this tells us that

T(t



, t) = T(t, t



)

†

= T(t, t



)

−1

. (7.41)

Thus the adjoint of a time development operator, which is the same as its inverse,

is obtained by interchanging its two arguments.

If one applies the dagger operation to (7.36), see (3.33), the result is

ψ

|=ψ



|T(t, t



)

†

=ψ



|T(t



, t). (7.42)

Consequently, the projectors [ψ

] and [ψ



] onto the rays containing |ψ

 and |ψ





are related by

[ψ

] =|ψ

ψ

|=T(t, t



)|ψ



ψ



|T(t



, t) = T(t, t



)[ψ



]T(t



, t). (7.43)

This formula can be generalized to the case in which P



is any projector onto some

subspace P



of the Hilbert space. Then P

deﬁned by

= T(t, t



T(t



, t) (7.44)

102 Unitary dynamics

is a projector onto a subspace P

with the property that if |ψ



 is any ket in P



its image T(t, t



)|ψ



 under the time translation operator lies in P

, and P

is com-

posed of kets of this form. That is to say, the same unitary dynamics which “moves”

one ket onto another through (7.36) “moves” subspaces in the manner indicated in

(7.44). The difference is that only a single T operator is needed to move kets, while

two are necessary in order to move a projector.

Since |ψ

 in (7.36) satisﬁes Schr

odinger’s equation (7.4), it follows that

∂T(t, t



)

∂t

= H(t)T(t, t



), (7.45)

where one can write H in place of H(t) if the Hamiltonian is independent of time.

There is a similar equation in which the ﬁrst argument of T(t, t



) is held ﬁxed,

−i

∂T(t, t



)

∂t



= T(t, t



)H(t



), (7.46)

obtained by taking the adjoint of (7.45) with the help of (7.41), and then inter-

changing t and t



. Given a time-independent orthonormal basis {|j}, (7.45) is

equivalent to a set of coupled ordinary differential equations for the matrix ele-

ments of T(t, t



∂

∂t

j|T(t, t



)|k=



j|H(t)|mm|T(t, t



)|k, (7.47)

and one can write down an analogous expression corresponding to (7.46).

Obtaining explicit forms for the time development operators is in general a very

difﬁcult task, since it is equivalent to integrating the Schr

odinger equation for all

possible initial conditions. However, if the Hamiltonian is independent of time,

one can write

T(t, t



) = e

−i(t−t



)H/



−iE

(t−t



e

|, (7.48)

where the E

and |e

 are the eigenvalues and eigenfunctions of H, (7.17). Thus

when the Hamiltonian is independent of time, T(t, t



) depends only on the differ-

ence t − t



of its two arguments.

7.4 Toy models

The unitary dynamics of most quantum systems is quite complicated and difﬁ-

cult to understand. Among the few exceptions are: trivial dynamics, in which

T(t



, t) = I independent of t and t



; a spin-half particle in a constant magnetic

ﬁeld with Hamiltonian (7.16); and the harmonic oscillator, which has a simple

time dependence because its energy levels have a uniform spacing, (7.22). Even a

7.4 Toy models 103

particle moving in one dimension in the potential V(x) = 0 represents a nontrivial

dynamical problem in quantum theory. Though one can write down closed-form

solutions, they tend to be a bit messy, especially in comparison with the simple

trajectory x = x

+ ( p

/m)t and p = p

in the classical phase space.

In order to gain some intuitive understanding of quantum dynamics, it is impor-

tant to have simple model systems whose properties can be worked out explicitly

with very little effort “on the back of an envelope”, but which allow more compli-

cated behavior than occurs in the case of a spin-half particle or a harmonic oscilla-

tor. We want to be able to discuss interference effects, measurements, radioactive

decay, and so forth. For this purpose toy models resembling the one introduced in

Sec. 2.5, where a particle can be located at one of a ﬁnite number of discrete sites,

turn out to be particularly useful. The key to obtaining simple dynamics in a toy

model is to make time (like space) a discrete variable. Thus we shall assume that

the time t takes on only integer values: −1, 0, 1, 2, etc. These could, in princi-

ple, be integer multiples of some very short interval of time, say 10

−50

seconds, so

discretization is not, by itself, much of a limitation (or simpliﬁcation).

Though it is not essential, in many cases one can assume that T(t, t



) depends

only on the time difference t − t



; this is the toy analog of a time-independent

Hamiltonian. Then one can write

T(t, t



) = T

t−t



, (7.49)

where the symbol T without any arguments will represent a unitary operator on

the (usually ﬁnite-dimensional) Hilbert space of the toy model. The strategy for

constructing a useful toy model is to make T a very simple operator, as in the

examples discussed below. Because t takes integer values, T(t, t



) is given by

integer powers of the operator T, and can be calculated by applying T several

times in a row. To be sure, these powers can be negative, but that is not so bad,

because we will be able to choose T in such a way that its inverse T

−1

= T

†

also a very simple operator.

As a ﬁrst example, consider the model introduced in Sec. 2.5 with a particle

located at one of M = M

+ M

+ 1 sites placed in a one-dimensional line and

labeled with an integer m in the interval

−M

≤ m ≤ M

, (7.50)

where M

and M

are large integers. This becomes a hopping model if the time

development operator T is set equal to the shift operator S deﬁned by

S|m=|m + 1, S|M

=|−M

. (7.51)

That is, during a single time step the particle hops one space to the right, but when

it comes to the maximum value of m it hops to the minimum value. Thus the